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14 TAYLOR'S FORMULA 195

11. Let I = [a, b] be a compact interval, and let M₀ be the vector space of real continuous
functions defined in I and such that, for any / e }a, b[, the limit

(L(/))« = Mm (f(t + fi)+f(t-h)-2f(t))/h*

Ji-»0,/igtO

exists in R. All real functions which are twice difFerentiable in I belong to M₀ .

(a) Let M be the vector subspace of M₀ consisting of functions /for which L (/) is
continuous in ]a, b[. Show that any function of /e M is twice dhTerentiable in ]a, b[
and that L(f) =/". (Use Problem 8(a) and 8(b), taking S = I, A = B = ]a, b[, and for
N the subspace of M consisting of functions /for which f(a) =/(&) = 0.)

(b) Show that the function f(t) = t cos(l/0 belongs to M₀ , although it is not dif-
ferentiable at t = 0.

12. What are the properties of functions with values in a Hilbert space which correspond
to the properties of real functions discussed in Problems 9(b), 10, and 11 ? (Cf. Section
8.5, Problem 6.)

13. Let / be an indefinitely differentiate mapping of a compact interval I = [a, b] c R
(with a ^ 0) into a Banach space.

(a) Show that, for any two integers p> q such that Q<p<q,

"f ll/»ll -. < °Z ll/wp) II -. + f ll/^toll r^r: dx

n=*p n\ n = p HI Ja (fl — 1/i

(For p^n<q, express /⁽ⁿ⁾(fl) using Taylor's formula at the point b.)

(b) Under the same conditions, show that

f* II/^(P)(*)II r^r. dx ^ f ||/">Wi| -?—- dx +

Ja (p— 1)! Ja (^~1)!

'^WII -

n = p Hi

(apply Taylor's formula to/^(p) and use Problem 4(c) of Section 8.11).

(c) Suppose /is indefinitely difFerentiable in the interval [a, +oo[ where a ^ 0, and
that for every integer n^Q, there is a finite number M_n such that ||/^<B)(x)||^//i! < M_wfor every x ^ a. Show that for b > a and n < q,

and for a ^ x ^ b

_m = fl \jn-qji

where the series and the integral are convergent (use Taylor's formula). Conclude that

one has

o _X9-i

and

^;c)l1

f ^ll/<P>(

(d) Show that, under the assumptions of (c), the function

(whose values are finite and 5*0, or +00) is increasing in [a, +oo[.



14 TAYLOR'S FORMULA 195 11. Let I = [a, b] be a compact interval, and let M₀ be the vector space of real continuous functions defined in I and such that, for any / e }a, b[, the limit (L(/))« = Mm (f(t + fi)+f(t-h)-2f(t))/h* Ji-»0,/igtO exists in R. All real functions which are twice difFerentiable in I belong to M₀ . (a) Let M be the vector subspace of M₀ consisting of functions /for which L (/) is continuous in ]a, b[. Show that any function of /e M is twice dhTerentiable in ]a, b[ and that L(f) =/". (Use Problem 8(a) and 8(b), taking S = I, A = B = ]a, b[, and for N the subspace of M consisting of functions /for which f(a) =/(&) = 0.) (b) Show that the function f(t) = t cos(l/0 belongs to M₀ , although it is not dif- ferentiable at t = 0. 12. What are the properties of functions with values in a Hilbert space which correspond to the properties of real functions discussed in Problems 9(b), 10, and 11 ? (Cf. Section 8.5, Problem 6.) 13. Let / be an indefinitely differentiate mapping of a compact interval I = [a, b] c R (with a ^ 0) into a Banach space. (a) Show that, for any two integers p> q such that Q<p<q, "f ll/»ll -. < °Z ll/wp) II -. + f ll/^toll r^r: dx *n=p n\ n = p HI Ja** (fl — 1/i (For p^n<q, express /⁽ⁿ⁾(fl) using Taylor's formula at the point b.) (b) Under the same conditions, show that

f* II/^(P)()II r^r. dx ^* f \|\|/">Wi\| -?—- dx + Ja (p— 1)! Ja (^~1)!		'^WII - n = p Hi

(apply Taylor's formula to/^(p) and use Problem 4(c) of Section 8.11). (c) Suppose /is indefinitely difFerentiable in the interval [a, +oo[ where a ^ 0, and that for every integer n^Q, there is a finite number M_n such that \|\|/^<B)(x)\|\|^//i! < M_wfor every x ^ a. Show that for b > a and n < q,

and for a ^ x ^ b

_m = fl *\jn-qji* where the series and the integral are convergent (use Taylor's formula). Conclude that one has o *_X9-i*

and


	^;c)l1
f ^ll/<P>( Ja

(d) Show that, under the assumptions of (c), the function

(whose values are finite and 5*0, or +00) is increasing in [a, +oo[.